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Find the number of functions of the form $g(x) = dx^3 + ex^2 + fx + g$ such that
\[g(x) g(-x) = g(x^3).\] | nvidia_math |
Two trains of some length and 200 m are 100 m apart. They start moving towards each other on parallel tracks, at speeds 54 kmph and 72 kmph. The trains meet after 2.856914303998537 minutes. What is the length of the first train? | tiger_math |
A boat goes 100 km downstream in 8 hours, and 75 Km upstream in 15 hours. The speed of the stream is?
Answer Choices: (A) 3 (B) 6.5 (C) 5.5 (D) 4 (E) 3.75 Let's program in Python in the response. | tiger_lab_math |
In a right triangle, one leg measures 6 units and the other leg measures 8 units. What is the length of the hypotenuse? | automath |
A car production factory is known to produce 20% defective cars. The cars are chosen randomly and tested. What is the probability that the 5th car tested is the first defective car found? Answer Choices: (A) 0.065 (B) 0.085 (C) 0.102 (D) 0.131 | tiger_lab_math |
In May, Alex collected 15 more seashells than Ben during their first beachcombing trip. However, Ben collected three times as many seashells as Alex during their second trip, while Alex's total number of seashells in the second trip was 30 less than the number he had collected in the first trip. If the total number of ... | nvidia_math |
There are 5 marbles in a bag - 2 are orange and 3 are blue. If two marbles are pulled from the bag, what is the probability W that at least one will be orange?
Answer Choices: (A) W=7/10 (B) W=3/5 (C) W=2/5 (D) W=3/10 (E) 1/10 | tiger_lab_math |
Given the fitted linear model for the relationship between height (in cm) and weight (in kg) stated as \(\widehat{weight} = -45.73 + 0.67 \times height\), what is the predicted weight (in kg) for a person who is 180 cm tall? Additionally, what is the interpretation of the coefficient for height in this model? | automath |
Emily owns 24 rare stamps in her collection. She decides to auction off a third of her stamps for $20 each. With the money earned from the auction, she buys new stamps for $8 each. How many stamps does Emily have after completing all her transactions? | nvidia_math |
Ethan got $75 for his birthday. He wants to buy three video games that cost $12 each and a gaming controller that costs $30. How much money will he have left? | nvidia_math |
Compute the sum of the geometric series $-1 -3-9-27 -81-243-729$. Let's program in Python in the response. | tiger_lab_math |
Given the vector \(\overrightarrow{a} = \langle 3, 2, -1 \rangle\), find a vector \(\overrightarrow{b}\) such that the scalar projection of \(\overrightarrow{b}\) onto \(\overrightarrow{a}\) is equal to 5. Specifically, express \(\overrightarrow{b}\) in vector form, and provide three different examples of such vectors. | automath |
Consider a rectangular sheet of metal measuring 10 cm by 12 cm. When constructing an open rectangular tray by cutting out squares of side length \( x \) from each corner, the volume \( V \) of the tray can be expressed as \( V = 4x(6 - x)(5 - x) \).
What values of \( x \) (in cm) make the volume \( V \) equal to 50 cm... | automath |
If an electron-positron pair is annihilated in a moving train and another pair is annihilated at rest on a platform, how does the frequency of the photons produced by the moving pair compare to the frequency of the photons produced by the stationary pair, assuming the train is moving at a significant fraction of the sp... | automath |
The area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is 8. Find the area of the parallelogram generated by the vectors $2 \mathbf{a} + 3 \mathbf{b}$ and $\mathbf{a} - 5 \mathbf{b}.$ | tiger_lab_math |
Brenda catches smallpox. She has 60 blisters on each arm and 80 blisters on the rest of her body. How many blisters does she have in total? Please write a program to solve it | tiger_lab_math |
Find the least common multiple of $3^5$ and $2^3 \cdot 5^2$.
To solve this problem, one needs to find the least common multiple (LCM) of two numbers, $3^5$ and $2^3 \cdot 5^2$. This requires a different approach than finding the greatest common divisor (GCD) in the original problem. One way to solve this problem is to... | nvidia_math |
A bookstore owner sold 30 copies of a bestseller novel for $20 each and 10 copies of a children's book for $15. The money earned was then used to pay the rent of $200 and to buy new bookshelves worth $100. How much money was left from the bookstore owner's earnings? | nvidia_math |
Simplify the following expression:
2x + 5 - 3x - 2 | tiger_lab_math |
Find the value of \( a \) such that \( 0 \leq a < 73 \) and \( a \equiv 9^{794} \mod 73 \). | automath |
The numbers 300 and 2000 are written in base 3. How many more digits does the base 3 representation of 2000 have than the base 3 representation of 300? | nvidia_math |
What is the smallest integer \( n \) such that the \( n \)th repunit number \( R_n \) is prime, and what is that prime repunit number? | automath |
There are 290 liters of oil in 24 cans. If 10 of the cans are holding x liters each, Each of the remaining cans is holding 15 oil. What is the value of unknown variable x? | tiger_math |
Solve for $x$:
$$x^3 - 2x^2 - 7x - 12 = (x + 3)(x^2 - x - 4).$$ | nvidia_math |
In January 2022, 3500 cases of influenza were reported in Florida. The state of Georgia had one-third the number of cases as Florida, while North Carolina had 200 fewer cases than Georgia. What is the total number of influenza cases reported in these three states? | nvidia_math |
A rocket starts from rest and moves upward from the surface of the Earth. For the first 10 seconds of its motion, the vertical acceleration of the rocket is given by \( a_y = (2.70 \, \text{m/s}^3)t \), where the +y-direction is upward.
What is the exact height of the rocket above the surface of the Earth at \( t = 1... | automath |
In the figure, $ABCD$ is a rectangle, $AZ=WC=6$ units, $AB=12$ units and the area of trapezoid $ZWCD$ is 120 square units. What is the area of triangle $BQW$?
[asy]
draw((0,0)--(12,0)--(12,20)--(0,20)--(0,0)--(12,20));
draw((0,14)--(12,6));
label("$A$",(0,20),W);
label("$Z$",(0,14),W);
label("$D$",(0,0),W);
label("$Q$... | nvidia_math |
Is it possible to have a set \( S \) such that its power set \( P(S) \) is countably infinite in Zermelo-Fraenkel set theory (ZF)? If not, what is the mathematical reasoning behind this conclusion? | automath |
Let
f(x) = 3x - 2
g(x) = 1/f(x)
h(x) = f(g(x))
What is the value of x for which h(x) = x? | nvidia_math |
Given the function \( g(t) = a \cos(\omega t) + b \sin(\omega t) \), express \( g(t) \) in terms of a complex phasor \( k e^{i(\omega t + \Phi)} \). What are the values of the constants \( k \) and \( \Phi \) in terms of \( a \) and \( b \)? Specifically, find \( k \) and \( \Phi \) such that \( g(t) = \text{Re}(k e^{i... | automath |
Let \( w = 7 + 2\sqrt{3} \). Compute the norm \( N(w) \) in \( \mathbb{Z}[\sqrt{3}] \) and determine whether \( w \) is a prime element of \( \mathbb{Z}[\sqrt{3}] \) by examining the irreducibility of \( w \) and the implications of its norm being a prime number in the ordinary integers. If \( w \) is prime, justify yo... | automath |
Consider the polynomial \( P(x) = x^3 + 12x^2 + 10ax + 1999 \). For which values of \( a \) does the polynomial definitely have at least one positive zero?
(a) \( a \geq 0 \)
(b) \( a > 0 \)
(c) \( a < 0 \)
(d) \( a \leq 0 \)
Provide a justification for your answer. | automath |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 5,$ $\|\mathbf{c}\| = 3,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$ | nvidia_math |
Consider a Trust Game between two players where Player 1 is given 10 tokens and must decide how many tokens to pass to Player 2. The tokens will be tripled in value once passed to Player 2, who will then have the option to return a portion of the tokens to Player 1. If Player 2 returns 4 tokens to Player 1, the final p... | tiger_lab_math |
Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? | tiger_lab_math |
What is the smallest integer that is greater than 1000 and is also a multiple of both 7 and 9? | nvidia_math |
Consider the following system of linear equations:
```
3x + 4y + 0z = 12
0x + 5y + 2z = 21
0x + 0y + 7z = 28
```
Express this system in matrix form, and solve the system using the sparse matrix method. Determine the values of x, y, and z. | tiger_lab_math |
If the angle $\theta$ in a triangle increases at a rate of $30$ degrees per second, what is the corresponding rate of change of $\sin \theta$ when $\theta$ is $60$ degrees? Provide your answer with the appropriate units. | automath |
Let $\mathbf{v}_0$ be a vector. This vector is projected onto $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ to obtain $\mathbf{v}_1$. Then, $\mathbf{v}_1$ is projected onto $\begin{pmatrix} 2 \\ Y \end{pmatrix}$ to obtain $\mathbf{v}_2$. If the matrix that transforms $\mathbf{v}_0$ to $\mathbf{v}_2$ is $\begin{pmatrix} 4 & 6 ... | tiger_math |
What is the 1000th 5-smooth number? | automath |
Given that the radius of circle A is 11 cm, circle B is 6 cm, and circle C is 5 cm, what is the total length of line segments AC and BC combined? | automath |
A constructor is using three water filling pumps, X, Y, and Z. Together, they can fill an empty tank in 9 hours. Pumps X and Y, working together, can fill the same tank in 10 hours. How many hours will pump Z take, working alone, to fill the same tank? | automath |
Ethan is reading a sci-fi book that has 360 pages. He read x pages on Saturday morning and another 10 pages at night. The next day he read twice the total pages as on Saturday. How many pages does he have left to read?
If we know the answer to the above question is 210, what is the value of unknown variable x? | tiger_math |
Given the probability density function of a continuous random variable \( Y \) defined as
\[
f_Y(y) = \frac{e^y}{(1+e^y)^2},
\]
which describes the standard logistic distribution, calculate the variance of \( Y \). Provide your answer as a numerical value. | automath |
A certain characteristic in a large population has a distribution that is symmetric about the mean m. 72 percent of the distribution lies within one standard deviation d of the mean. If the shelf’s average life is 7.6 years and the standard deviation is 2.6 years, what percent of the distribution has more than 10.2 yea... | tiger_lab_math |
Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$. Note that $\bar{z}$ denotes the conjugate of $z$. | tiger_math |
A 1165 cm long rope is cut into 154 pieces. 150 of the pieces are equally sized and the remaining pieces are 100mm each. Find the length of each of the equal pieces in millimeters. | tiger_lab_math |
Let \( w \) be a character vector defined as follows:
\[
w \leftarrow c("Adam", "Bob", "Charlie", "David", "Emma", "Fiona", "George")
\]
If you retrieve entries from vector \( w \) using the index \( w[c(1, 4, 7)] \), what will be the resulting vector? | automath |
Acme T-Shirt Company charges a $\$50$ set-up fee plus $\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme? | tiger_math |
"statement 1": Ratio between Rahul and Deepak is less than 7 : 3 , After 6 Years Rahul age will be 26 years, "statement 2" :Ratio between Rahul and Deepak is 4 : 3 , After 6 Years Rahul age will be 26 years, "options: " Entailment or contradiction or neutral? Let's program in Python in the response. | tiger_lab_math |
The function f(x) is defined as follows:
f(x) = x^2 + 1, if 0 ≤ x < 2
f(x) = 3, if 2 ≤ x < 4
f(x) = x - 1, if 4 ≤ x ≤ 6
Determine the sum of the x-coordinates of all points where f(x) intersects with the line y = 2x. | nvidia_math |
Let \(T : V \to W\) be a linear transformation from an \(n\)-dimensional vector space \(V\) to an \(m\)-dimensional vector space \(W\). If the rank of the transformation \(T\) is \(r\), what is the dimension of the null space of \(T\)? Express your answer in terms of \(n\) and \(r\). | automath |
If the diameter of the large pulley is 2 inches, the diameter of the small pulley is 1 inch, and the center-to-center distance between the two pulleys is 3 inches, what is the length of the V belt? | automath |
Given a flow rate of flue gas of 20 cubic meters per minute and a concentration of nitrogen oxides (NOx) of 10 parts per million by volume, calculate the mass flow rate of NOx in grams per hour. Use the molar mass of NOx, which is 46 kg/kmol, and assume that the molar volume of a gas at 0 °C and 101.325 kPa is 22.414 m... | automath |
Is the statement \(((p \to q) \land (q \to r)) \to (p \to r)\) a tautology in propositional logic, and if so, how can it be proved using truth tables or logical inference rules? | automath |
Blake gave some of his money to his friend, Connie. Connie used all the money she had been given to buy some land. In one year, the land tripled in value. So, she sold the land and gave half of all the money she got from the sale of the land to Blake. If Connie gave Blake $30,000, how much money, in dollars, did Bl... | tiger_math |
A metal detector generates a uniform magnetic field perpendicular to its surface and is held at an angle of 15.0° to the ground. Below the surface lies a silver bracelet consisting of 6 circular loops, each with a radius of 5.00 cm, and the loops are parallel to the ground. If the magnetic field increases at a constant... | automath |
Does every separable metrizable space \( X \) possess at least one compactification \( bX \) such that the remainder \( bX \setminus X \) is also separable and metrizable? | automath |
Consider a rectangular channel with a cross-sectional area of flow \( A_c = 120 \, \text{m}^2 \) and a mean depth of \( h_m = 3 \, \text{m} \). What is the surface width \( T \) of the water in the channel? | automath |
Oliver buys 2 printers and 5 scanners at the electronics store. Each printer costs $50. The total cost of Oliver's items is $220. If each scanner is the same price, what is the cost of 3 scanners? | nvidia_math |
Factor $36-9x^2$. | nvidia_math |
a train 300 m long can cross an electric pole in 30 sec and then find the speed of the train ? | tiger_math |
What is the maximum integer \( m \) such that the ring of integers modulo \( m \), denoted \( \mathbb{Z}_m \), is an integral domain? | automath |
The ratio of the amount of money Cara, Janet, and Jerry have is 4:5:6. If the total amount of money they have is $75, calculate the loss Cara and Janet will make if they combine their money to buy oranges and sell them at 80% of the buying price. | tiger_lab_math |
if the number of white balls in a box are 6 times the black balls , which of the below is the total number of balls , if there are 8 black balls in the box ? Let's program in Python in the response. | tiger_lab_math |
When the length of a simple pendulum is increased by 22 cm, resulting in a 20% change in its period, what is the original length of the pendulum? | automath |
If there are Six periods in each working day of a school, In how many ways can one arrange 5 subjects such that each subject is allowed at least one period?
Answer Choices: (A) 2388 (B) 3600 (C) 2791 (D) 1817 (E) 1911 | tiger_lab_math |
The difference of two numbers is 1311. On dividing the larger number by the smaller, we get a certain quotient and 11 as the remainder. The larger number is 1430. What is the quotient when dividing the larger number by the smaller number? | tiger_math |
If x is equal to the sum of the integers from 60 to 80, inclusive, and y is the number of even integers from 60 to 80, inclusive, what is the value of y? | tiger_math |
Mr. Thompson received holiday cards from 25% of his students. 2/5 of these cards included a bookstore voucher worth $15. If he collected a total of $90 in vouchers, how many students did he have in his class? | nvidia_math |
in the standard formulation of a flavored drink the ratio by volume of flavoring to corn syrup to water is 1 : 12 : 30 . in the ` ` sport ' ' formulation , the ratio of flavoring to corn syrup is 3 times as great as in the standard formulation , and the ratio of flavoring to water is half that of the standard formulati... | tiger_lab_math |
If $\cot \alpha = 4$ and $\cot \beta = 3,$ then find $\cot (\alpha - \beta).$ | tiger_math |
Emily wants to order a cake from a bakery for her birthday party. The large cake costs $25.00 and is cut into 12 slices. The first filling costs $3.00, the next 2 fillings cost $2.00 each, and the rest of the fillings cost $1.00. If she orders a large cake with chocolate, vanilla, strawberry, caramel, nuts, and sprinkl... | nvidia_math |
Calculate the surface area generated when the arc of the curve \( y = \cosh(x) \) between \( x = 0 \) and \( x = \ln(a) \) is rotated about the x-axis. Express your answer in terms of \( a \). | automath |
Ken likes to bike when it's raining and can cycle 30 miles in 20 minutes during this time. However, when it's snowing Ken can't stand the cold and can only cycle 10 miles in 20 minutes. If it rains 3 times and snows 4 times in one week, how many miles did Ken reach if he cycles 1 hour a day? | tiger_lab_math |
Determine the range of the function $g(x) = \arcsin x + \arccos x - \arctan x.$ All functions are in radians. | nvidia_math |
Evaluate the integral
\[
I = \int_0^1 \frac{4x-5}{\sqrt{3+2x-x^2}} \, dx
\]
and express the result in the form
\[
I = \frac{a \sqrt{3} + b - \pi}{6},
\]
where \(a\) and \(b\) are constants. What are the values of \(a\) and \(b\)? | automath |
Let \( f(x) = x^3 - 27x \).
1. (a) Determine the intervals on which the function \( f(x) \) is increasing and decreasing. What are the critical points of \( f(x) \) and how do they affect the behavior of the function on the real line?
2. (b) Consider the series \( \sum_{n=1}^{\infty} \frac{2n+1}{n^2+2n+1} \) and \( ... | automath |
Given the joint probability density function \( f(x,y) = c \frac{2^{(x+y)}}{x! y!} \) for discrete random variables \( X \) and \( Y \) defined for \( x = 0, 1, 2, \ldots \) and \( y = 0, 1, 2, \ldots \), find the value of \( c \) such that the total probability sums to 1. | automath |
Find the coefficient of \( x^{12} \) in the expansion of \( (2x + 3x^3)^{8n} \). | automath |
Suppose there is a certain number that leaves a remainder of 2 when divided by 5 and also leaves a remainder of 3 when divided by 7. What is the smallest positive integer that satisfies this condition? | tiger_lab_math |
Let \( P(x) = (x-1)(x-2)(x-3) \). For how many distinct polynomials \( Q(x) \) of degree 2 exist such that there is a polynomial \( R(x) \) of degree 3 that satisfies the equation \( P(Q(x)) = P(x) \cdot R(x) \)? | automath |
In a convex hexagon, the sum of the measures of all interior angles is 720 degrees. If the measures of three consecutive angles are equal and the measures of the remaining angles are equal, what is the measure of each of the larger angles, in degrees? | nvidia_math |
In a store, there are boxes of crayons in 3 colors. The store has 30 boxes of blue crayons. The number of boxes of red crayons is twice the number of blue crayons. Boxes of yellow crayons make up 15% of the total number of red crayons. How many boxes of crayons are of a different color if the total number of crayon box... | nvidia_math |
A rectangular prism has a total surface area of $1200$ square meters, and the sum of the lengths of all its edges is $440$ meters. If the lengths of all three sides are positive integers, find the volume of the prism. | nvidia_math |
What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$? | nvidia_math |
Jane is looking at a tree. At the same time, the tree and Jane are casting shadows - the tree of 10 meters and Jane of 0.5 meters. Jane is 1.5 meters tall. How tall is the tree? | tiger_lab_math |
Veronica, who is the team captain of their school football team, carried x yellow balls and 33 brown balls to the field for the team's matchday practice. What percentage of the balls that she carried were yellow?
If we know the answer to the above question is 45, what is the value of unknown variable x? | tiger_math |
Given the expressions for the volume charge density of a neutral helium atom, where the nucleus has a charge density of \(\rho_n(\vec{r})=2q\delta^3(\vec{r})\) and the electron cloud has a density of \(\rho_e(\vec{r})=-\frac{8q}{\pi {a_0}^3}e^{-4r/{a_0}}\), calculate the electric field \(\vec{E}\) at a distance \(r\) f... | automath |
Find the sum of the two solutions to the equation $\displaystyle\sqrt[3]{4 - \frac{x^2}{3}} = -2$. | tiger_math |
Jason is climbing a telephone pole next to a tree his friend Matt is climbing. Matt can climb 6 feet/minute and Jason can climb 12 feet per minute. After a certain amount of time, Jason is 42 feet higher than Matt. How many minutes have they been climbing? | tiger_math |
the compound and the simple interests on a certain sum at the same rate of interest for two years are rs . 11730 and rs . 10200 respectively . find the sum ? Let's write a program. | tiger_lab_math |
A group of 7 friends are planning to sit in a row at a movie theater. What is the probability that no two friends who are sitting next to each other have the same favorite color (red, blue, or green), given that 3 friends like red, 2 friends like blue, and 2 friends like green? | nvidia_math |
Given that the constant value of the unnormalized circulation \( \Gamma \) about a cylindrical airfoil at steady state can be represented by the equation \( \Gamma = \frac{Q \cdot U}{V} \) where \( Q \) is the circulation parameter related to the tangential air speed \( V \), and \( U \) is a constant proportional to t... | automath |
Let \( f: M \to N \) be a continuous function, with \( K \) being a compact subset of \( M \). Assume that for every open set \( B \subseteq N \), the preimage \( f^{-1}(B) \) is open in \( M \). If \( A_i \) is an open cover of \( f(K) \), explain why there exists a finite subcover of \( K \) and specifically how this... | automath |
John buys 1/2 gallon jugs of cold brew coffee every x days. He drinks 2 cups of coffee a day. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
A standard deck of 52 cards has 4 suits, each with 13 cards. If two cards are drawn randomly from the deck, what is the probability that the product of the numbers on the cards will be a multiple of 5? Express your answer as a common fraction. | nvidia_math |
Consider the expression \( \frac{256^{4/7}}{2^{4/7}} \). Simplify this expression and find its exact value. What result do you obtain? | automath |
Thelma is planning to buy tickets to a concert for her friends. There are two types of tickets available: VIP tickets for $\$$50 each and regular tickets for $\$$20 each. She wants to buy a total of 15 tickets and has a budget of $\$$600. However, the concert organizers have a promotion where for every 3 VIP tickets pu... | nvidia_math |
Find the sum of the two solutions to the equation $\displaystyle\sqrt[3]{4 - \frac{x^2}{3}} = -2$. | tiger_math |
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